Divergence of the logarithm of a unimodular monodromy matrix near the edges of the Brillouin zone

Abstract

A first-order differential system with matrix of periodic coefficients $Q(y)=Q(y+T)$ is studied for time-harmonic elastic waves in a unidirectionally periodic medium, for which the monodromy matrix $M(\omega )$ implies a propagator of the wave field over a period. The main interest in the matrix logarithm $\ln M(\omega )$ is due to the fact that it yields the 'effective' matrix $Q_{eff}(\omega )$ of the dynamic-homogenization method. For the typical case of a unimodular matrix $M(\omega )$ ($\det M=1$), it is established that the components of $\ln M(\omega )$ diverge as $(\omega -{\omega }_0{)}^{-1/2}$ with $\omega \to {\omega }_0,$ where ${\omega }_0$ is the set of frequencies of the passband/stopband crossovers at the edges of the first Brillouin zone. The divergence disappears for a homogeneous medium. Mathematical and physical aspects of this observation are discussed. Explicit analytical examples of $Q_{eff}(\omega )$ and of its diverging asymptotics at $\omega \to {\omega }_0$ are provided for a model of scalar waves in a two-component periodic structure. The case of high contrast due to stiff/soft layers or soft springs is elaborated. Special attention in this case is given to the asymptotics of $Q_{eff}(\omega )$ near the first stopband that occurs at the Brillouin-zone edge at arbitrary low frequency. The link to the quasi-static asymptotics of the same $Q_{eff}(\omega )$ near the point $\omega =0$ is also elucidated.

20 pages, 1 figure

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